production costs - university of ghana ... costs social costs of production, refer to costs to...

PRODUCTION COSTSIn this section we introduce production costs into the analysis of the firm. So far, our emphasis has been on the production process without any consideration of costs.

However, production activities do involve costs – implicit and explicit.

But cost is a rather complicated concept. It is a term that is open to more interpretations. Note for example the difference between consumers’ cost and producers.

1Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION COSTSSocial Costs of production, refer to costs to society when its resources are employed to make a given commodity. Since economic resources are limited, when resources are used to produce a certain product, less can be produced of some other product that can be made with those resources.

Private costs, are in contrast with social costs. Private costs are defined to be costs to the individual firm or producer. Take case of pollution!


PRODUCTION COSTS

Explicit costs include the ordinary items that

an accountant would include as the firms

Expenses, e.g., payroll, payments for raw

materials, etc.

Implicit costs (alternative costs or opportunity

cost) include opportunity costs of resources

owned and used by the firm’s owner. This type

is often omitted in calculating the firm’s costs.

These costs generally come under private costs


PRODUCTION COSTS

The economic cost of any activity is the value

of the best forsaken alternative.

The firm in order to attract the resources or

“factors” necessary to engage in production,

must pay resource owners amounts sufficient to

induce them to sacrifice their best alternatives

(whether employment or leisure).

But it is important to note that in production,

cost and price are different.


PRODUCTION COSTSAs we shall see later, costs are important in determining a firm’s optimum profit position.

They are also the basis for a firm’s supply curve.

Having noted all these, it is worth stating the goal of the firm. Economists usually assume that the firm maximises profit, which is defined as the difference between revenue and cost.

This section on costs assumes that the firm under analysis is a competitive or “price-taking firm”.


PRODUCTION COSTS

Distinguishing between short-run and long-run costs

This is related to the same concept in production theory.

In the short-run some costs are fixed, whilst in the long-run they become variable. This is the fundamental difference between the two.

Nevertheless, the distinction is a matter of degree.


PRODUCTION COSTSDistinguishing between short-run and long-

run costs

The longer the run contemplated, the greater the range of costs regarded as variable rather than fixed.

Hence, if a firm is not committed to any outlays, it is in the long-run. In the long-run all options are open!

The situation changes to the short-run once a commitment to some factor of production has been undertaken.


PRODUCTION COSTS

Distinguishing between short-run and

long-run costs

Thus, the short-run is characterised by

fixed and variable costs.

In the long-run, all costs are variable

since all costs depend on the volume of

output.


PRODUCTION COSTS

Total, Average and Marginal Costs

The total cost can be regarded as the sum, taken

over all resources employed, of factor prices times

factor quantities.

In other words, it is the sum of all costs incurred

by the firm to produce a given level of output.

From the total cost, two other measures emerge:

average and marginal costs.


PRODUCTION COSTS


The average cost (AC) is defined as the cost

per unit of output. Formally, this is defined as:

Where TC is total cost and Q is output

Q

TCAC


PRODUCTION COSTS


The marginal cost (AC) is defined as the

change in total cost resulting from a unit change

in output. Formally, this is defined as:

Where TC is total cost and Q is output

Q

TCMC


PRODUCTION COSTS


Some important conclusions are worth making!

We have noted earlier that in the short-run, the

firm faces both fixed and variable costs. But as the

firm alters its output, only the variable costs

change (why?).

The marginal cost that a firm experiences as it

expands output from given fixed resources are

entirely due to its variable costs.12Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION COSTS


This therefore leads to a major conclusion!

Decisions about output are based

entirely on marginal costs; fixed

costs are totally irrelevant to any

output decisions.


PRODUCTION COSTS


. But based on our understanding from

production, we know that cost is a multivariable

function, that is, it is determined by many

factors.

Thus, in the short-run,

),,,(

KPTXfC f


PRODUCTION COSTS


Where C = total costs; X = output; Pf=

prices of factors; T = technology; and

= fixed factors.

In the long-run

K

),,( fPTXfC


PRODUCTION COSTS


We have earlier made the assumption that the

firm is a competitive firm.

Thus, it seeks to be efficient in production,

aiming to produce at the minimum cost of

production for any given output level, Q, when

factor prices are Pf .

If we also assume that firms are price takers in

the factor markets, then Pf is fixed.


PRODUCTION COSTS


Thus, we can write our cost function as

dependent on output, Q, alone. This can be written

as

Hence total costs can be written as

)(QcC

FCQcC )(


PRODUCTION COSTS


The marginal cost function can be written as

Q

FC

Q

Qc

Q

QcQMC v

)()()(

VCQ

QcQMC v

)()(


PRODUCTION COSTS


But note that the marginal cost measures the rate of change, hence we can define the marginal cost function as

Refresh your memory on the relationship between MC, AVC and AC.

dQ

Qcd

dQ

dTCQMC

)]([)(


PRODUCTION COSTS


PRODUCTION COSTS


The marginal cost curve, average variable

cost curve and average total cost curves are

generally U-shaped.

The U-shape in the short run is attributed to

increasing and diminishing returns from a

fixed-size plant, because the size of the plant is

not variable in the short run.


PRODUCTION COSTS


The marginal cost and average cost curves are

related:

When MC exceeds AC, average cost must

be rising

When MC is less than AC, average cost

must be falling

This relationship explains why marginal cost

curves always intersect average cost curves at

the minimum of the average cost curve.


PRODUCTION DECISION

Optimal Input Combinations

How will a competitive firm combine inputs to produce a given quantity of output?

As a first approximation, we assume that firms are out-and-out profit maximizers; that is to maximize the difference between revenue (R) and (economic) costs (C) incurred.


PRODUCTION DECISION


Thus, our typical firm seeks to maximize profits;

The assumption of profit maximization implies

that a firm will seek to minimize the costs of

producing a given output or seek to maximize

the output derived from a given level of cost.

CR max


PRODUCTION DECISION


Also remember the assumption of perfect

factor markets, such that firms are price takers

in the input markets.

Thus, if we suppose there are two inputs,

labour (L) and capital (K), then what

combinations of L and K should the firm

choose?


PRODUCTION DECISION


If the wage rate (w) is the cost of labour and

the rental rate (r) is the cost of capital, then the

total cost outlay, C, is given by:

Lr

w

r

CK

rKwLC


PRODUCTION DECISION


Thus, the various combinations of L and K

that can be purchased, given input prices and

total outlays can be represented by a straight

line. K

LC/w

C/rSlope = (w/r)

0


PRODUCTION DECISION


A family of Isocost lines can be illustrated

belowK

L0

C(3)

C(2)C(1)

C(3) > C(2) > C(1)


Maximization of output for given cost

L

K

0

100

200

300R

L*

K*


PRODUCTION DECISION

Optimization Condition

The firm maximizes output at point R, by

choosing L* and K* of labour and capital

respectively .

At point R, the isoquant is tangent to the

isoquant. Thus,

r

w

MP

MPMRTS

K

LKL ,


PRODUCTION DECISION


It follows that the optimal combination of inputs

is where.

Or

r

w

MP

MP

K

L

r

MP

w

MP KL


PRODUCTION DECISION


More generally, a firm will choose an input

combination such that.

Where MPa, MPb, ... ..., MPn are the marginal

products of inputs, a, b, ..., n, and Pa, Pb, ... ..., Pn

are input prices.

n

n

b

b

a

a

P

MP

P

MP

P

MP .......


Minimization of Cost for a Given

Output Level

L

K

0

400

Z

L*

K*


PRODUCTION DECISION


To minimize the cost of producing the output level, Q=400, the firm chooses point Z. Here too, the firm must equate the MRTS to the ratio of input prices

r

w

MP

MPMRTS

K

LKL ,


PRODUCTION DECISIONConstrained Optimization

The firm’s decision, as with the case of consumers, can be represented as a constrained optimization problem.

We first consider the case of constrained output maximization.

Thus, given Q = f(K, L) and C = rK + wL, we can set up the Lagrangian function:

)(),( CwLrKLKfZ


PRODUCTION DECISION

Constrained Optimization

Z symbolises the Lagrangian function. Here

is an undetermined Lagrange multiplier

Also note that another formulation of the

Lagrangian function is:

)(),( wLrKCLKfZ

0


PRODUCTION DECISION


We set the first-order conditions, which are to

set to the partial derivatives of K, L, and λ equal to

zero.

)1(0

rf

K

Zk

)2(0

wf

L

Zl


PRODUCTION DECISION


From (1) and (2): moving the price terms to the

right and dividing (2) by (1):

)3(0

CwLrK

Z

)4(r

w

f

f

k

l


PRODUCTION DECISION


From (4) the first order conditions state that the

ratio of marginal products must be equated with

their price ratios.

Solving (1) and (2) for λ, yields:

)5(w

f

r

f lk


PRODUCTION DECISION


(5) states that the contribution to output

of the last money outlay expended on each

input must equal λ.

The multiplier, λ, is the derivate of

output with respect.


PRODUCTION DECISION

Constrained Cost Minimization

The firm may desire to minimize the cost of

producing a prescribed level of output.

As with our earlier analysis, we form the

Lagrangian function, and set the partial

derivatives to zero for K, L, and λ.

),(( 0 LKfQwLrKZ


PRODUCTION DECISION


An alternative formulation of the Lagrangian

function is:

In what follows, we set the various partial

derivatives to zero and obtain the optimal

conditions for input combination in production

)),(( 0QLKfwLrKZ


PRODUCTION DECISION


)'1(0

kfr

K

Z

)'2(0

lfw

L

Z

)'3(0),(0

LKfQ

Z


PRODUCTION DECISION


Since r and w are both positive, moving the

price terms and dividing (2’) by (1’), we obtain:

The first order conditions for the minimization

of cost subject to an output constraint are similar

to those for the maximization of output subject to

a cost constraint.

MRTSf

f

r

w

k

l


PRODUCTION DECISION

Second-Order Conditions

We shall leave out the details of second-order

conditions. However, more generally, for output

maximization subject to a given cost, and for cost

minimization subject to a given output level, the

slope of the marginal product curves for the two

inputs must be negative.

or: 0;02

2

2

2

L

Q

K

Q 0;0 llkk ff


PRODUCTION DECISION


The second-order conditions ensure that we

are satisfied that the isoquants are convex to

the origin.

If the isoquant is concave, then we have a

corner solution.


PRODUCTION DECISION


e

e1

e2 L

K

0

Q=230


PRODUCTION DECISION


In the diagram above, output, Q=230, can be

produced at points e, e1 and e2.

This indicates different costs of producing the same level of output.

The lowest cost point is given by e2.


PRODUCTION DECISIONProfit Maximization: A Formal Analysis

A firm is usually free to vary the levels of both cost and output, with the ultimate objective being to maximize profits rather than a solution to a constrained maximum or constrained minimum problems.

The firm we have been analysing is assumed to operate in a competitive market. Its total revenue is given by the number of units of Q sold multiplied by the fixed unit price it receives.


PRODUCTION DECISION

Profit Maximization: A Formal Analysis

The firm’s profit is thus defined as:

Given Q = f(K, L) and C = rK + wL, the firm’s profit function is given by

П = Pf(K, L) – rK - wL

Profit is a function of K and L and is thus maximised with respect to these variables.

CQP .


PRODUCTION DECISION


Differentiating the profit function with

respect to capital and labour, gives:

Moving input price items to the right, we

have:

)6(0 rPfkk

)7(0 wPfll


PRODUCTION DECISION


Thus, Pfk and Pfl are the values of the

marginal product of K and L respectively, and

they represent the rate at which the firm’s

revenue would increase with further increases

in K or L.

rPfk wPfl


PRODUCTION DECISION


Profit maximization requires that each input be utilised up to the point at which the value of its marginal product equals its price.

Profits can be increases as long as Pfk > rand Pfl > w.

That is, as long as the addition to revenue from employing an additional unit of input exceeds its cost.


PRODUCTION DECISION


The second-order conditions for profit maximization require that:

These suggest that profits must be decreasing with respect to further increases in L and K.

02

2

llPf

L0

2

2

kkPf

K


PRODUCTION DECISION


Because P > 0, this suggests that the marginal

product of both L and K must be decreasing.

The conditions for first- and second-order

profit maximization require that the isoquant be

strictly concave in the neighbourhood of the a

point at which the first-order conditions are

satisfied with non-negative levels of inputs (K

and L).55Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION DECISION

Optimal Expansion Path in the Short-Run

In the short-run, K is fixed, the firm is therefore

forced to expand along a straight line parallel to

the horizontal axis.

With prices of factors constant, the firm does not

maximise profits in the short-run, due to the

constraint of given capital.

The optimal path would be along OA, but the

firm can only expand along in the short run.__

KK


PRODUCTION DECISION

Optimal Expansion Path in the Short-Run

0

_

K_

K

A

K

L57Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION DECISION

Optimal Expansion Path in the Long-Run

In the long-run all factors are variable.

Output can therefore be expanded without

limitation.

As is always the case, the firm’s objective of

profit maximization, this means it chooses the

least-cost combination of inputs, which is

represented by the points of tangency between

the isocosts curves and isoquants.


PRODUCTION DECISION


0

E

K

L

15012080


PRODUCTION DECISIONOptimal Expansion Path in the Long-Run

The expansion path indicates how, as output rate changes (but input prices remain fixed),the quantity of each input changes.

If the production function is homogeneous, the expansion path will be a straight line through the origin, whose slope depends on the ratio of factor prices.

With only two inputs in production, it is also easy to derive the long-run cost function from the expansion path.


PRODUCTION DECISION


It is also worth noting that the maximum

profit-input combination lies on the expansion

path.

Given Pfl = w and Pfk = r, we note that this is a

special case of the constrained output

maximization discussed earlier.


PRODUCTION DECISION


That is, along the long-run expansion path, the

condition is satisfied.

Also the implies that profit is also maximised,

that is,

r

w

f

f

k

l

r

w

Pf

Pf

k

l


PRODUCTION DECISION

Input Demand Functions

The firm’s input demands are derived from the

underlying demand for the goods and services it

produces.

Thus, the firm’s input demand functions are

obtained by solving the first-order conditions for

profit maximization for L and K, as functions of

input prices and output price.


PRODUCTION DECISION


Therefore, more generally, given a production

function of the Cobb-Douglas form, we can obtain

the firm’s input demand functions

1:0,, KALQ

rKwLKPAL


PRODUCTION DECISION


Solving for L and K, we obtain:

01 wKALPl

01 rKALPk

),,(;),,( ** prwfKprwfL

0;0;0 prw fff65Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

production costs - university of ghana ... costs social costs of production, refer to costs to...

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